Mathmatic Modeliing f Cmmunicatin Networks

8/6/2019 Mathmatic Modeliing f Cmmunicatin Networks

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Mathematical modelling and analysis

of communication networks:

Transient characteristics of traffic processes

and models for end-to-end delay and delay-

jitter

by

Olav sterb

Telenor Research and Development

Submitted to NTNU-Department of Telematics

in partial fulfilment of the requirements

for the degree of Dr. Philos.


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To Silje, Sunniva and

Hkon Olav


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If in addition the nodes are identical i.e. the convolution consists of the waiting times of a

fixed numbers of identical M/G/1 queues, the evaluation may be substantially simplified. It

turns out the convolutions may be found by taking some partial derivatives with respect to

the load parameter. The same technique may be generalised in various directions for in-

stance it is possible to extend the result to the case with two groups of queues where the

queues in each group are identical.

For M/D/1 queues with identical service times we find explicit closed form results for the

convolutions. We also generalize this result to consider two groups of M/D/1 queues hav-

ing different service times, and this is a particularly interesting case since it may be used as

model for end-to-end delay also including access links with low capacity. Similar results are

also found for end-to-end queueing models with priority.

A different approach is obtained by assuming a slotted model. The main idea is to capture

the disturbance of a packet stream as it passes through a series of multiplexers. Even though

the output process from a multiplexer is non-renewal, we get the distribution between two

consecutive departures, and approximate the process with a renewal stream. This stream is

then feed into the next multiplexer (together with other crossing traffic). In this way we ob-

tain recursive relations for the jitter and the end-to-end delay.


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0Preface

The writing of this thesis, which is the result of several years of research, has finally cometo its end. The work has been carried out during the years 1998-2003 partly as ordinary re-

search projects, but also a lot of spare time, late evenings and weekends have been spent at

office and at home to finalise the thesis.

During the year 1998-1999 I was privileged to work full time doing basic research, and I

would like to thank may employer Telenor R&D giving me the opportunity to start the re-

search activity. Without that year of research there would definitive not been any thesis. I

would like to thank Terje Ormhaug and Harald Pettersen for their support when I applied

for sabbatical year and Mette Rhne for encouraging me to do so, at that time when she

was finalising her own doctoral thesis and I was her assistant supervisor.

There are several other persons to whom I am grateful. Among them I would like to men-

tion my superior Nils Flaarnning for encouraging me to finalise the dissertation and notimposing me with any extra tasks during the last year. Inge Svinnset, my project manag-

er and (train travelling companion), for his tolerance and for making room for plenty of my

research in his projects. Ralph Lorentzen, for helping me with the proof of theorem A.1.

Thor Gunnar Eskedal for having time for talks both professional and personal. Last but not

least, Terje Jensen for always being interested in discussing technical questions, giving con-

structive criticism to various proposals, and for reading the manuscript.

With NTNU Department of Telematics I have benefit from good co-operation for a number

of years, and I would particular thank Prof. Peder J. Emstad and Prof. Bjarne E. Helvik for

several stimulating discussions.

I would also like to thank my family for their support and patience during these years, and I

would dedicate the thesis to my children Silje, Sunniva, and Hkon Olav.

Olav sterb

Strmmen ,18th August. 2003


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4.2 Gaussian traffic models ....................................................................................344.3 The n-point approximation for Gaussian processes .........................................374.3.1 n-point approximation of the distribution of excess times and first passage

times for a stationary Gaussian process .......................................................384.3.2 Joint n-point approximation of the distribution of excess times and

excess volumes and first passage times and the corresponding volumefor a stationary Gaussian process ................................................................42

4.3.3 Some numerical examples ...........................................................................464.4 Distribution of the first passage times and the corresponding volumes for

the Ornstein-Uhlenbeck process .......................................................................534.4.1 First passage time distribution for the Ornstein-Uhlenbeck process ...........544.4.2 Joint distribution of the first passage time and the corresponding volume

for the Ornstein-Uhlenbeck process ............................................................614.4.3 Some numerical examples ...........................................................................69

5 Some results on excess times and excess volume for semi-Markov processes ....755.1 Introduction ......................................................................................................755.2 Some general properties for semi-Markov processes .......................................765.3 The first two moments for the excess times and excess volumes for

semi-Markov processes ....................................................................................795.3.1 The case when the time spent in a state is independent of the next state ....815.3.2 Markov processes ........................................................................................815.3.3 Birth-death semi-Markov processes ............................................................815.4 Distribution of the excess times and excess volumes .......................................835.4.1 General formulae for semi-Markov processes .............................................83

5.4.2 Spectral decomposition of the distributions ................................................875.4.3 Birth-death semi-Markov processes ............................................................895.4.4 Markov processes ........................................................................................935.5 Some numerical examples ................................................................................96

Part-II Models for calculating end-to-end delay and delay-jitter in

packet networks..................................................................................................105

6 Introduction .........................................................................................................1076.1 Addressing the QoS ........................................................................................1086.2 Performance issues .........................................................................................1096.2.1 IP-multiplexing for low capacity links ......................................................1096.2.2 Addressing the end-to-end queueing delay ...............................................109

6.3 Reference configuration .................................................................................1106.4 The organisation of PART-II of the thesis .....................................................111

7 Convolution of a given number of waiting times of M/G/1queues havingidentical service time distributions .....................................................................113

7.1 Some preliminary considerations ...................................................................1137.2 Convolution of waiting times in M/G/1 queues all having identically

distributed service times .................................................................................1147.3 Convolution of waiting times in M/G/1 queues having different

service times ...................................................................................................118


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7.3.1 Convolution of the waiting time distribution for a given number ofM/D/1 queues all with equal service times ................................................119

7.3.2 Convolution of waiting times in M/D/1 queues having differentservice times ..............................................................................................120

7.3.3 Asymptotic approximations .......................................................................1247.4 Some numerical examples ..............................................................................1277.4.1 Cases with identical nodes that are equally loaded ...................................1287.4.2 Numerical examples including one or more low capacity access links ....1357.5 Concluding remarks .......................................................................................140

8 Convolution of a given number of waiting times of M/G/1 non-preemptivepriority queues having identical service time distributions ................................141

8.1 Some preliminary considerations ...................................................................1418.2 Exact results when all the nodes are identical ................................................1438.2.1 Deterministic service times for low priority packets .................................1448.2.2 Exponentially distributed service times for low priority packets ..............1468.3 Approximative methods .................................................................................1478.4 Examples ........................................................................................................1498.5 Concluding remarks .......................................................................................156

9 Discrete time queueing models ...........................................................................1579.1 Introduction ....................................................................................................1579.2 A discrete time queueing model with a renewal foreground and a batch

background stream as input ............................................................................1579.2.1 Transient queueing analysis .......................................................................159

9.2.2 Stationary queue length distribution ..........................................................1629.3 Delay and delay jitter for the FS ....................................................................1649.3.1 The FS packet is always served first when it arrives together with

BS packets .................................................................................................1649.3.2 The FS packet and possible BS packets arriving in the same slot are

served at random ........................................................................................1659.3.3 The FS packet is always served last when it arrives together with

BS packets .................................................................................................1669.3.4 Inter-departure time, jitter and queueing delay distributions ....................1669.3.5 Some variants of the inter-departure time and jitter z-transforms .............1689.4 Heavy and light traffic analysis ......................................................................1709.5 End-to-end delay and jitter evaluation for a stream traversing a series of

queueing nodes ...............................................................................................172

9.6 Some comments on the numerical procedure to calculate the end-to-enddelay and jitter ................................................................................................174

9.7 Some numerical examples ..............................................................................1759.7.1 End-to-end delay ........................................................................................1779.7.2 Evolution of the jitter .................................................................................1799.8 Concluding remarks .......................................................................................184

Bibliography ........................................................................................................185


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Appendix A Crossing intensities and the joint probability of the excess volume andthe excess time that starts in (0,dt1) and ends in (t,t+dt2) .............................191

Appendix B Some important properties for multinormal integrals ...................................197

Appendix C The sign of the real part of the poles of the Laplace transformsin section 5.4 .................................................................................................225

Appendix D Alternative expressions for the Laplace transforms in section 5.4.3 ............227

Appendix E Asymptotics of the excess distributions and first passage times for theU-O process ....................................................................................................237

Appendix F Some technical details in chapter 7 and chapter 8 ........................................247

Appendix G Some technical details in chapter 9 ..............................................................259


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Acronyms

ADSL Asynchronous Digital Subscriber Line

ATM Asynchronous Transfer Mode

a.s. almost sure

BS Background stream

CAC Call Acceptance Control

CDF Complementary Distribution Function

DF Distribution Function

DiffServ Differentiated Services

DSL Digital Subscriber Line

F First

FCFS First Come First Serve

FIFO First In First Out

FS Foreground stream

HOL Head Of LineIntServ Integrated Services

IP Internet Protocol

L Last

LD Large Deviation

LST Laplace-Stieltjes Transform

MPLS Multi Protocol Label Switching

O-U Orstein-Uhlenbeck

NA Normal Approximation

PDF Probability Density Function

PHB Per Hop Behaviour

QoS Quality of Service

R RandomRT Real Time

SLA Service Level Agreement

TCP Transmission Control Protocol

STM Synchronous Transport Module

UAA Uniform Asymptotic Approximation

VDSL Very high speed Digital Subscriber Line


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1

1Overall introduction to the thesis

1.1 Background and motivation

There is an increasing need to address and understand some fundamental teletraffic issues in

todays and forthcoming communication network, where the current trend is a change-over

towards heterogeneous network types where the end-to-end communication may involve

more than one operator and the QoS (Quality of Service) provisioning is not currently satis-

factorily solved. New switching techniques emerge, capacity increases, and new services are

introduced, causing a steady growth in the traffic. The competition in the telecom market is

hard, the revenue is squeezed, so the slogan throwing bandwidth at the problems, will not

be a winning strategy for the operators. On the other hand too scarce network resources

could result in degradation in the quality of the services offered, leading to discontented

customers with the subsequent consequences that may cause. So there is a strong need for

network optimization and performance modelling.

By the emerging of IP (Internet Protocol) multiservice networks a lot of new teletraffic

challenges emerge. Among those we would particularly mention:

- traffic models for different services or flows and also models for aggregation of

flows,

- differentiation between classes of services, i.e. scheduling and buffer management,

- traffic control, i.e. SLA (Service Level Agreement), CAC (Call Acceptance Control)

and policing,

- QoS provisioning end-to-end,

- dimensioning models.

Thus, the need for performance and dimensioning modelling of todays communication net-works is sustained and is definitive not of less importance than for former types of net-

works.

1.2 Main achievements

In this thesis we have focused on a few of the issues mentioned above; namely models to

describe transient behaviour of rate processes and models to get the end-to-end delay in

packet networks. As such, we are confident that these are important models; providing in-

sight into important aspects of networking, as congestion periods and information loss and


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the delay and delay variation for a packet flow. The main objective have been to obtain an-

alytical results based on mathematical modelling, but where we always take the applied

viewpoint, where the goals have been to provide models that are numerical feasible and

provide numerical examples that are interesting from the perspective of network perform-

ance. In the analysis we have derived results on the basis of several applied mathematical

fields such as: probability theory and queueing analysis, asymptotic expansions of integrals,

differential equations and a few results from real analysis. We may summarise the main

achievements obtained as:

- Give some general results on level crossing and excess distribution for stationary rate

processes.

- Suggest approximations of the excess times and excess volumes distributions for

general stationary Gaussian process.

- Give expansions and asymptotic formulae for the PDF (Probability Density Func-

tion) and CDF (Complementary Distribution Function) of first passage times and

corresponding volumes for the Orstein-Uhlenbeck (O-U) process.

- Give some general results for level crossing and excess distributions for semi-Mark-

ov processes where we for more specific models as birth-death process obtain the

LST (Laplace-Stieltjes Transforms) recursively due to the special structure of corre-

sponding generator matrices.

- Give an effective method to obtain the PDF and DF (Distribution Function) of the

convolution of a given number of waiting times of identical M/G/1 queues.

- Extend the result on convolutions to cover cases where not all the service times may

be identically distributed, and also to cover HOL (Head Of Line) priority queueing.

- Provide a slotted queueing model and using generating function techniques to obtain

the output distribution of particular packet stream and apply this model recursively to

obtain both end-to-end delay and the evolution of the jitter for a deterministic pack-

et stream through a series of nodes.

All the numerical examples are obtained by Mathematica programming except for the ex-

amples in chapter 5 which is obtained by FORTRAN routines. Many of the numerical re-

sults are checked against similar (but different) models and also against asymptotic expan-

sions.

1.3 Overall organization of the thesis

This thesis is divided into two parts, Part I and Part II, but where each chapter more or less

is self-contained with an introductory chapter. A quite large number of more technical de-

tails are put in separate appendices. The use of symbols throughout the thesis is not strin-

gent, for instance is the symbol used to denote a rate process in Part I while in Part II the

same symbol is used as a symbol of the number of packet arrivals from a background

stream during a slot. However, within each chapter the notation should be consistent.

Part I is devoted to find methods to describe transient behaviour of traffic processes, where

the main emphasis is placed on the description and analysis of excess periods and excess

B


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volumes of quite general stochastic processes, and the organisation is described in section

2.3.

Part II provides models to obtain traffic dependent end-to-end queueing delay and give

methods to calculate evolution of the jitter through out a network. The organisation of Part

II is given in section 6.4.


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PART-I

1Some results on level crossing, excessdistributions and first passage times for

stationary rate processes


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2

2Introduction

Traffic models are needed as an input for dimensioning and performance evaluation of tele-communication systems. They will also be important in the process of designing and struc-

turing networks. In this context we consider fresh traffic, i.e. before it has entered some

network elements where it could be disturbed by other traffic streams. By fresh traffic we

here shall mean the raw bit rate generated from various traffic sources. However, we must

to some extent also include the effect from different protocol layers adding on overhead and

framing the bit stream into packets. We shall also consider traffic models that are a super-

posed or an enveloped traffic stream, formed as a collection of individual sources, where

the gross bit rate is taken as the sum the instantaneous bit rates from the sources as if there

where no other constraints or limitations, e.g. buffers, capacity etc.

2.1 Traffic modelling and scaling phenomena

The presence of scaling phenomena in some type of data traffic is well documented in theliterature. The area of analysis, modelling and characterization of traffic in communication

networks has quickly evolving since the seminal paper by Leland et al. [Lela93] in 1993.

This paper showed that network traffic in many cases has properties characterized by long-

range dependence and variability at a wide range of time scales, and it introduced the no-

tion of self-similarity to communication networks. Later on, these properties have been

shown to hold also for a much wider range of experimental environments [Paxs95],

[Crov96].

The evidence of traffic being long range dependent has certain implication on the behav-

iour of the autocorrelation function for large arguments, resulting in a power law behaviour

with exponent between zero and unity, where the exponent is expressed in terms of the

Hurst-parameter by . We therefore aim to look at models that have long-range de-pendence. On the other hand the behaviour of the autocorrelation for small arguments will

determine the behaviour of the processes at micro-level (at least for processes that have

continuous sample paths) indicating that one possibly should use different models on the

micro level than on the macro level.

2.2 Transient characteristic of traffic processes

By assuming that traffic change on different time scales, the transient characteristics of the

traffic processes could be an important measure to describe periods of congestion on a com-

2H 2


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munication link (by assuming that the relaxation time for the link buffer is of order less

than the typical length of an excess period). Moreover, the corresponding excess volume

will represent lost information during such periods. For traffic model acting on a (virtual)

link (of fixed capacity), it is of great interest to be able to answer to some of the following

questions:

- How often will the excess periods occur?

- What is the distribution of the length of the excess periods?

- What is the distribution of the corresponding excess volumes?

Although the results obtained are of rather general nature, they give some rather fundamen-

tal insight into transient characteristic of traffic processes. The aim has been to provide

some results concerning the duration of excess periods and the corresponding excess vol-

umes. It turns out that the up- and down-crossing rate is an important measure, and will an-

swer the first question above provided that the limit is finite. The distributions of the length

of an excess period may then be expressed it terms of some excess probabilities that is re-

lated to the minimum of the process in the time interval considered. Similar relations for the

excess volume is harder to obtain and requires the joint probability of the arrived volume

and the minimum of the process in the same time interval. Unless for some very special

models the exact excess probability is difficult to find expressions for and therefore some

approximations will be needed to get results that are numerical feasible.

2.3 The organisation of PART-I of the thesis

PART-I of the thesis is a collection of three chapters. The main focus through these chap-

ters is the use of level crossing to describe transient phenomena as excess times and excess

volumes for bit rate processes. As mentioned these periods may represent periods of con-

gestion for bufferless multiplexing.

Chapter 3 deals with fundamental questions concerning level crossings for stationary sto-

chastic processes where we discuss some of the basic properties. It is known that level

crossing is a rather tricky matter, and put strong limitations of the class of processes, espe-

cially for processes with continuous sample paths. The chapter is logically divided into two

parts where we in the first part give some fundamental results and the only assumption is

that the process is stationary, while we in the second part consider processes that are contin-

uous in time and space and we describe a method that makes it possible to also include the

excess volumes into the analysis.

In the first part we define the crossing rate and deduce that if this rate is finite it is given as

negative derivative of the excess probability. Then we discuss the relation between the ex-

cess probabilities and the distribution of the excess periods. In the second part we consider

processes that are continuous in time and space. The claim of having finite crossing rate for

such processes will put rather strong implications on the behaviour of the autocorrelation

near the origin. For processes with continuous sample paths we also give relations between

some joint excess probabilities and the joint distribution of the first passage times and cor-

responding volumes. Similar relations are also found for the joint distribution of the excess

times and corresponding excess volumes, but they are more tricky to obtain.


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In chapter 4 we consider Gaussian traffic models. In the first part we propose an approx-

imative method to obtain the distributions of excess times and excess volumes. The main

idea is to approximate the excess probabilities by multinormal integrals. Based on these ide-

as we express both first passage times and corresponding volumes and the excess times and

corresponding excess volumes in terms of multinormal integrals. In a separate appendix

(Appendix B) we have given many interesting properties of such type of integrals, among

them the result that makes it possible to calculate the multinormal integrals by calculating a

multiple integral with only half the dimension.

In the second part we consider the Ornstein-Uhlenbeck (O-U) process. The main motiva-

tion was firstly because the O-U process is a special case of Gaussian process and could

therefore be used as a test process for the proposed approximations. Secondly, since the O-

U process may be obtained as a limit of a large numbers of ON/OFF sources (with expo-

nential distributed ON- and OFF- times) the results is important in its own. For the O-U

process we have given the Laplace transforms for the first passage times and the corre-

sponding volumes. These Laplace transforms are inverted by locating the residues yielding

infinite series. Asymptotic expansions for small arguments are also found.

In a series of numerical examples we first tested the approximation by applying multinor-

mal integral of dimension five or six with the exact first passage time distributions for the

O-U process. Unfortunately the correspondence was not as good as we hoped, however, the

proposed approximation seems to yield an upper bound for the distribution functions. In a

second series of examples we chose a process with typical long rang dependence. We con-

clude that the corresponding (approximative) excess time seems to have long tails.

In chapter 5 we consider the case when the bit rate process is a semi-Markov process. This

type of process is not limited by the claim on the behaviour of autocorrelation function near

the origin as found for processes with continuous sample paths.

The main results obtained are general expressions for the Laplace transforms and distribu-

tion functions of the excess times and the excess volumes in terms of the generator matri-

ces. For birth-death semi-Markov processes the generator matrices simplify and the trans-

forms may be found recursively. For ordinary Markov processes the excess distributions

may be obtained by finding the eigenvalues to the corresponding rate matrices, and finally

for birth-death processes these eigenvalues may be effectively found by applying the meth-

od of bisection.


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3

3Some transient characteristics of trafficanalysed by methods of level crossing and

excess distributions

3.1 Introduction

If statistical multiplexing is allowed in broadband networks it may happen that the load

form the ongoing communications (or connections) may exceed the capacity of a particular

link. Due to statistical fluctuations this situations may occur even though the network is

well dimensioned. This may lead to periods with excessive information loss and thereby

possible degradation of the QoS. The time scale of such variations may typical be that of an

ON/OFF activity period of a frame duration for a video source. At this level the discrete na-

ture of the transmission e.g. packets (or cells) are negligible, and we consider a more or less

continuous bit stream with different characteristics. Thus rather than considering eventswhere arrivals of packets or cells occur we take the fluid approach where we observe a con-

tinuous bit stream representing the traffic under consideration.

In the literature statistical fluctuations and level crossing initially appeared in the field of

statistical communication theory and analog signal processing. The first result on level

crossing is due to Rice [Rice45], and goes actually back to 1936 where he gave the classic

formula on the average rate of level crossing for Gaussian processes. In his context the

main focus was the ability to detect levels of a signal that was influenced by random noise.

(See also [Rice48].) Along with other authors in the same field there exists a quite large

numbers of papers considering level crossings for Gausian processes, however the main fo-

cus has been the crossing of the zero level, whereas we are mainly interesting in crossing of

levels having small probabilities. In some quite early papers by McFadden [McFa56] and

[McFa58] some quite general results are given for axis crossing. Similar results where also

the distributions between successive zeros are discussed are found in the papers [Long58]

and [Long62] but with the assumption of a Gaussian process. In the book of Leadbetter et

al. [Lead83] a quite large number of results on level crossing are given mainly for Gaussi-

na processes, but also some basic results concerning the crossing intensity for general proc-

esses that have continuous sample paths are given. Also in several other textbooks as

[Lars79b], [Papo65] and [Midd60] the topic of axis crossing for Gaussian processes have

been treated.


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Our aim in this chapter is not to give a complete mathematical treatment of the different

topics, but rather take the perspective of an engineer and give some general results that are

important for performance measures characterising rate processes. A more thorough mathe-

matical treatment will require techniques that are beyond the scope of this thesis.

The usual way of characterising dependencies in stochastic processes is to introduce the

second order statistics; that is the covariance or autocorrelation function. The level-crossing

description introduced below will be more appropriate to get the performance measures

needed for instance considering bufferless multiplexing. This fact will be evident when we

derive formulas for the second order moments of the excess volumes that may quite easily

be obtained by using the level-crossing description. Also the new achievements obtained by

using large deviations seem to fit well into this description.

3.2 Some general results concerning the excess times and excess volumesfor stationary stochastic processes

By taking a general starting point we let be a (non-negative) stochastic process repre-

senting the instantaneous bit rate (load) on communication link. The main assumption we

put on the bit rate process is that it is stationary (in the strict sense), which means that any

group has the same distribution as for all choices of .

It follows that the behaviour of the process is independent of the staring point of the obser-

vations (which we in most cases choose to be ). We shall also limit ourselves to con-

sider only time continuous processes. For discrete time processes a similar development is

possible but such models will not be discussed in this thesis.

In the following we let denote theMean bit rate and be the

corresponding Variance and we let denote the Autocorrelation func-

tion of the process. It turns out that the behaviour of the Autocorrelation function near the

origin will provide the necessary information to determine whether the up and down cross-

ing intensities are finite, and therefore determine when the description below is fruitful or

not. (See for instance the introductory textbooks of stochastic processes [Cinl75] [Cox70]

[Fell68a] [Fell68b].)

Assume that we for a given level (link capacity) may identify up and down crossing in-

stants and such that in the interval and in the

interval (see figure 3.1). The possible up and down crossings intervals

and will describe periods of congestion and non-congestion for buff-

erless multiplexing, or period of buffer filling or buffer emptying in a fluid queue.

Bt{ }

Bt1 Btn, ,{ } B t+ 1 B t+ n, ,{ }

t 0=

m E B0[ ]= 2

E B02

[ ] m2

=

t( )E B0Bt[ ] m

2

2

----------------------------------=

C

Uk{ } Dk{ } Bt C 0> UkDk,( ) Bt C 0

Dk U, k 1+( )

UkDk,( ) Dk U, k 1+( )


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By assuming that it is possible to identify the random sequences and of the up

and down crossing instants, we define the excess times and excess volumes:

and (3.1)

and similar periods of normal load and corresponding volumes:

and (3.2)

For a loss system will describe the amount of information lost during a congestion peri-

od but for the fluid model and describe the net increase and decrease in buffer con-

tent during the intervals and . We also define the total volume arriv-

ing during two consecutive down crossings as .

The main contribution in this chapter will be to describe a general framework to get the dis-tributions (and moments) of the length of these intervals and the corresponding volumes. If

it is possible to obtain these distributions they will give interesting performance measures

such as the length of overload periods and the time between them. By considering the ex-

cess volumes it is possible to estimate the information loss for bufferless multiplexing and

especially the losses in the period of overload.

Figure 3.1: Definition of up and down crossing instance.

U1 D1 U2 D2 U3

time t

BtCapacity level C

Uk{ } Dk{ }

Tk Dk= Uk Ak B C( ) d

Uk

Dk

=

Sk U= k 1+ Dk Vk C B ( ) d

Dk

Uk 1+

=

Ak

Ak Vk

UkDk,( ) Dk U, k 1+( )

Wk B d

Dk

Dk 1+

=


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3.2.1 Some general remarks on level crossing

It turns out that one need to be some careful when considering up and down crossing for

stochastic processes. This is specially seen in processes having continuous sample paths

where the term up and down crossing can be defined slight different. For instance in the

book of M. R. Leadbetter et al. [Lead83] it is defined both the term up crossing and strict

up crossing where a so called non strict up crossing is allowing for infinite many up cross-

ings in a small interval. Since we consider processes that have both continuous and piece-

wise continuous sample paths we shall take the following definition:

A function (which we assume to be piece-wise continuous) is said to have an up cross-

ing of the level at a point if for some and every , then for all

in the interval and for some in the interval . (In a simi-

lar way we also define down crossing.)

In the following we shall derive some quite general expressions for the excess distribution

on the basis of the basic knowledge of the bit rate process . It turns out that the cross-

ing intensities may be expressed through the functions (excess probabilities):

and (3.3)

(3.4)

The functions (and ) are the probabilities that the process either is above(or below) the level (and does not crosses that level) in an interval of length . It

will be convenient to approximate the process by a sequence taking the val-

ue of at points ( ) and let being linear between such

points. (With this type of partition we obtain the th from the th by halving each

interval and therefore doubling the number of points.) We may now approximate

by the corresponding -point approximation and

we define:

(3.5)

Similar we also define the -point approximation for the maximum

and we define

(3.6)

f t( )

C t t0= 0> 0> t( ) C

t t0 t0,( ) t( ) C> t t0 t0 +,( )

Bt{ }

C t( ) P Inf 0 t,( ) B C>{ }=

C t( ) P Sup 0 t,( ) B C{ }=

C

t( ) C

t( )

C t

Bt{ } Btn

{ }

Bt{ } tin i

2n

-----t= i 0 1 2n

, , ,= Btn

{ }

n 1+( ) n

mt Inf 0 t,( ) B= 2n

mtn

Min0 i 2

n

Btin=

Cn t( ) P Min

0 i 2n

B

tin C>{ } P B

t0n C B

t1n C B

tnn C>, ,>,>{ }= =

2n

t

nMax0 i n Bti

n=

Mt Su p 0 t,( ) B=

Cn

t( ) P Max0 i 2

n

Btin C{ } P B

t0n C B

t1n C B

tnn C, ,,{ }= =


29/278


30/278

- 16 -

(3.13)

It turns out that the instantaneous crossing rate will play an important part in the effort to

find expressions for the different excess distributions. Before we go further to find the dif-

ferent moments and distributions of the different excess times we shall first show the fol-

lowing important results which relate crossing rate to the derivative of the excess func-

tions (and ) at :

(3.14)

To prove (3.14) we start with the obvious inequality

implying that

. The last inequality shows that if then

when .

Next we assume that is finite. Then we also have

for (with

equality for ) giving . By writing out the difference

as follows we find:

Each term in the last sum is obviously bounded by

(where the equality is due to the

assumption of stationarity of the stochastic process). We therefore have

for (with equality for ). Combining the two

inequalities above gives:

C C t( )t 0lim C

t

2n

-----

n lim= =

C

C t( ) C t( ) t 0=

C C 0( ) C 0( )= Cn

0( ) Cn

0( )== =

C t( ) P B0 C Bt C>,>{ } P B0 C>{ } P B0 C> Bt{ }=

C t( ) C 0( )

t----------------------------------- C t( ) C t( )

C t( ) C 0( )

t----------------------------------- t 0

C C t( )t 0lim=

Cn

t( ) P B0 C Bt C>,>{ } P B0 C>{ } P B0 C> Bt{ }= n 0 1 2 , , ,=

n 0=Cn t( ) C 0( )

t----------------------------------- C t( )

C 0( ) Cn

t( )

C 0( ) Cn

t( ) P B0 C>{ } P Bt0n C B

t1n C B

tnn C>, ,>,>{ }= =

P Bt0n C B

t1n C B

ti 1n C>, ,>,>

P Bt0n C B

t1n C B

tin C>, ,>,>

i 1=

2

P Bt0nC B

t1n C B

ti 1n C B

tin C,>, ,>,>

i 1=

2

=

P Bti 1n C Bti

n C,>{ } P B0 C> B t2

n-----

t

2n-----C

t

2n----- = =

C 0( ) Cn

t( )

t--------------------------------- C

t

2n

----- n 0 1 2 , , ,= n 0=


31/278

- 17 -

(with equality for ) (3.15)

If we now fix and let in (3.15) then we get:

(3.16)

and the result (3.14) follows now by letting in (3.15) and (3.16).

As a consequence of the inequality we get the following bounds on the function and

(for small ):

and (3.17)

(3.18)

The proof for is similar and is therefore omitted, but the corresponding inequalities

(3.15) and (3.16) yield by replacing with and with and further

the bounds (3.17) and (3.18) will read:

and (3.19)

(3.20)

(We should mention here that when we derive (3.14) by (3.15) and (3.16) we only assume

that subintervals are of equal lengths, which means the results also yield for the case where

the we divide the interval into subinterval of equal lengths )

It is also of interest to find the probability of having more than one crossing (of the level

) in an interval of length when is finite. We let denote this probability and

find the following result:

If the crossing intensity is finite then or as .

To show this result we define the probability of having an even number of crossings in an

interval of length by . Since in this case either both the starting point and the end

point are above or below the level we must have

Ct

2n

----- C

nt( ) C 0( )

t----------------------------------- C t( ) n 0=

t n

CC t( ) C 0( )

t----------------------------------- C t( )

t 0

C t( )

Cn

t( ) t

P B0 C>{ } tC C t( ) P B0 C>{ } tC t( )

P B0 C>{ } tCt

2n

----- C

nt( ) P B0 C>{ } tC t( )

C t( )

C t( ) C t( ) C t( ) C t( )

P B0 C{ } tC C t( ) P B0 C{ } tC t( )

P B0 C{ } tCt

2n

----- C

n t( ) P B0 C{ } tC t( )

t

n---

C t C C t( )

CC t( )

t--------------

t 0lim 0= C t( ) o t( )= t 0

t Ceven

t( )

C


32/278

- 18 -

. Similar we also define the probability of hav-

ing an odd number (greater than one) of crossings in an interval of length by . To

have an odd number (greater than one) of crossings in an interval of length we must have

an even number of crossing in the interval , a single crossing in and no

crossing in . Therefore will be bounded by the integral of , that

is we have for some . Then by (3.15)

(where we have equality for ) and (3.16) we get

for . If we let

the result follows.

By applying the nice property above we may neglect the probability of having more than

one level crossing in a small interval (of say length since we will have

) and this will heavily simplify the derivation of the different excess

distributions below where we always shall assume that the crossing intensity is finite.

3.2.2 Distribution and moments of the excess times

In this section we shall discuss a general framework to get the excess time distributions

and for a general stationary stochastic process. Sometimes we also want to get the time

to the first down crossing (first passage time) conditioning on the bit rate process (when we

are inside an excess period). We therefore also define to be the time to the first down

crossing for the process when we start the observation in an excess period with

.

It turns out that the first passage time may be expressed in terms of what we call envelope

probabilities (or excess probabilities) defined by:

for , and (3.21)

for , (3.22)

and where we also denote the corresponding densities and

.

Ceven

t( ) C0

t( ) C t( ) C0

t( ) C t( )+=

t Cod d

t( )

t

0 ,( ) d+,( )

d+ t,( ) Cod d

t( ) Ceven

t( )

Cod d

t( ) Ceven

( )

0=

t

d tCeven

t1( )= 0 t1 t

n 0=

0C t( )

t--------------

Ceven

t( ) Cod d

t( )+

t--------------------------------------------- 2 C C t( )( ) 2t C C t1( )( )+= 0 t1 t

t 0

dt

C dt( ) dt o dt ( )=

Tk

Sk

Tx

Bt{ }

B0 x=

FC

x y t , ,( ) P Bt

y> Inf 0 t,( )

B

C B0

x=>,{ }= x C y C

GCx y t , ,( ) P Bt y Su p 0 t,( ) B C B0 x=,{ }= x C y C

Cx y t , ,( ) y

FCx y t , ,( )=

gCx y t , ,( ) y

GCx y t , ,( )=


33/278


34/278

- 20 -

Conditioning on this event and observing that the event

we may express

the excess time distribution as

; and by rewriting the event

above and using the assumption of having a stationary process the last expression may be

rewritten as:

(3.26)

provided that is finite. Hence, these results state that the CDF of an excess period for a

general stationary stochastic process is given as the normalized derivative of the excess

probability (3.24). The result (3.26) requires that crossing intensity (or the derivative at

time ) exists and is finite. Unfortunately this is not the case for some rather interest-

ing classes of continuous processes. For the Ornstein-Uhlenbeck (U-O) process the deriva-

tive will become arbitrary large for small values of so the limit (3.26) will be-

come zero for all . This is in accordance with the well known rapid oscillations for the

Wiener process and the O-U process described in many textbooks for stochastic processes

[Cox70], [Karl66]. (We shall discuss the O-U process in section 4.4.)

An alternative to consider the excess time distribution defined above (which does not al-ways exists) we may consider a somewhat simpler variable taken to be the first passage

time given that the bit rate process is above the level . The corresponding CDF may

be found by integrating (3.23) by the conditional distribution of given that :

(3.27)

To obtain the CDF of the time distribution of the normal load period; that is the distribu-

tion of we proceed as for the overload case, and we get:

(3.28)

On the basis of (3.26) and (3.28) it is straight forward to find the first two moments of the

excess times:

and and (3.29)

B0 C Inf dt t,( ) B C>,{ } Inf dt t,( ) B C>{ } Inf 0 t,( ) B C>{ }=

P Tk t>( ) P B0 C Inf dt t,( ) B C B0 C Bdt C>,>,( )dt 0lim=

FTkt( ) P Tk t>( )=

C t dt( ) C t( )

C 0( ) C dt( )---------------------------------------------

dt 0lim=

C t( )

C-------------------=

C

C

t 0=

C t( ) t

t

TC C

B0 B0 C>

P TC t>( ) FC

x C=

x C t , ,( )d x( )

P B0 C>( )-------------------------=

C t( )

P B0 C>( )-------------------------=

Sk

FSkt( ) P Sk t>( )=

C t dt( ) C t( )

C 0( ) C dt( )-------------------------------------------

dt 0lim=

C t( )

C------------------=

E Tk[ ]P B0 C>( )

C-------------------------= E Sk[ ]

P B0 C( )

C-------------------------=


35/278

- 21 -

and . (3.30)

In words it is possible to express the mean excess times as the stationary probability that

the bit rate is above (or below) the capacity limit divided by the rate at which the process

cross that level.

We may also obtain the expected time between two consecutive up or down crossings:

(3.31)

Thus, the mean excess times above (or below) a given capacity level could as well be de-

rived by direct arguments; as the portion of time the process is above (or below) a given

level decided by the up or down crossing rate.

3.2.3 Moments of the excess volumes

By taking the expectation of the stochastic integrals (3.2) we may express the mean values

of the excess volumes as:

and (3.32)

(3.33)

Based on the results above we may estimate the overall information loss as the ratio

of the mean excess volume and the mean traffic volume in a cycle:

(3.34)

and the information loss in an overload period as the ratio of the mean excess vol-

ume and the mean traffic volume in an excess period:

E Tk2

[ ]

2 C

0

t( )dt

C--------------------------= E Sk

2[ ]

2 C

0

t( )dt

C-------------------------=

E Uk 1+ Uk[ ] E Sk[ ] E Tk[ ]+1

C------= =

E Ak[ ] E Tk[ ]E B0 C B0 C>[ ]

P B0 x>( ) xd

C

C----------------------------------= =

E Vk[ ] E Sk[ ]E C B 0 B0 C[ ]

P B0 x( ) xd

0

C

C---------------------------------= =

ploss

ploss

P B0 x>( ) xd

C

m----------------------------------

qloss


36/278

- 22 -

(3.35)

We observe that the crossing rate is not included in the estimates (3.34) and (3.35) and they

may as well be used for processes where crossing rate does not exist. To justify this we

may consider a sampled version of the process (where we take the sampled version linear

between samples). If we choose a sample interval of length the corresponding crossing

rate is and by choosing small (but not ) we will get the estimates (3.34)

and (3.35).To obtain the second order moments of the excess volumes we define the conditional covar-

iances

(3.36)

(3.37)

First we calculate the conditional moment . After some manipulations we find:

(3.38)

By the theorem of double expectation (see for instance [Cinl75]) we get from (3.38):

(3.39)

By proceeding in the same way for we finally obtain:

(3.40)

qloss

P B0 x>( ) xd

C

mP B0 C>( )----------------------------------

C ( ) 0=

C t( ) E Bt C( ) B0 C( )1 Inf 0 t,( ) B C>{ }[ ] x C( ) y C( ) x( )fCx y t , ,( ) xd yd

y C=

x C=

= =

C t( ) E C B t( ) C B 0( )1 Su p 0 t,( ) B C{ }[ ] C x( ) C y( ) x( )gCx y t , ,( ) xd yd

y 0=

C

x 0=

C

= =

E Ak2

Tk[ ]

E Ak2

Tk[ ] 2 Tk ( )C ( )

C ( )--------------- d

0=

Tk

=

E Ak2

[ ] E E Ak2

Tk[ ][ ]=

2 C t( ) td

t 0=

C------------------------------=

Vk

E Vk2

[ ]

2 C t( ) td

t 0=

C-------------------------------=


37/278

- 23 -

3.3 Further results for bit rate processes that are continuous in time andspace

The rest of this chapter will be devoted to processes that have continuous sample paths and

are absolute continuous (which means that the PDF for the process exists and is a continu-

ous function). In the literature there exists sufficient criteria for a stationary process having

continuous sample paths. We refer to the textbook of H. J. Larson and B. O. Shubert

[Lars79b] where it is stated that if it possible to find constants such that

for (3.41)

where and then the process is sample path continuous. If the process is Gaus-

sian one can relax the demand on to have . In the following we shall assume that

these criteria are fulfilled to be sure that the corresponding processes are sample path con-

tinuous.

3.3.1 Up and down crossing intensity

For absolute continuous processes it is possible to write the crossings intensity as an inte-

gral. To do this we define which is the differential process scaled by

(when is small the scaled differential process will be close to the derivative of provid-

ed that the derivative exists).

By conditioning on we may write the up and down crossing intensity as:

and (3.42)

(3.43)

where is the PDF of . If the functions and

are uniform continuous with respect to for all

then one may take the limit under the integral sign in (3.42) and (3.43) giving:

(3.44)

1 t( ) at

0 t l 1>

0>

dB tBt B0

t-----------------= 1 t

t Bt

B0

Cuc

t( )1

t---P B0 C Bt B0 C ty=( ) C ty( ) yd

y 0=

= =

Cdc

t( )1

t---P B0 C> Bt{ } P d Bt y> B0 C ty+=( ) C ty+( ) yd

y 0=

= =

z( ) B0 F1ty z,( ) P dB t y> B0 z=( ) z( )=

F2ty z,( ) P d Bt y> B0 z=( ) z( )= z y

t 0

C1

2--- C

uct( ) C

dct( )+( )

t 0lim

C( )2

------------ P dB t y> B0 C=( ) P+ d Bt y> B0 C=( ) yd( )

y 0=

t 0lim= =

C( )2

------------ E dB t B0 C=[ ]t 0lim=


38/278


39/278

- 25 -

and furthermore by the monotone conver-

gence theorem [Royd68] is finite a.s. and further

(3.48)

Of course (3.46) put a quite strict limitation on the class of autocorrelation functions for

which it is meaningful to use the notion of up and down crossing intervals even though the

processes may have continuous sample paths. Thus, the requirement of having continuoussample paths combined by requirements of having finite up and down crossing intensities

will limit the class of processes to those having autocorrelation function of type (3.46). This

is in strict contrast to processes with jumps, for instance stationary Markov processes where

the up and down crossing intensities will exists even though the autocorrelation function is

on the form for small where is a positive constant. (Processes

with jumps are discussed in detail in chapter 5 in this thesis.)

As a side results, by applying the inequality above, we find the integral over the up or down

crossing intensity is bounded by that is:

(3.49)

Summarising the discussion above we have shown that sufficient condition that the up and

down crossing intensities exist and are finite a.s. is that the autocorrelation is on the form

(3.46). Whether this also is a necessary condition will not tried to be answered in this the-

sis, however, for a Gaussian processes this is the case. As a side result of the discussion we

obtain a bound on the integral of the crossing rates and provide a measure of the variability

in the process that is proportional with the standard deviation and the square root of the sec-

ond order derivative of the autocorrelation (at the origin) with negative sign.

3.3.2 Joint distribution of the first passage time and the corresponding volumeThe general formulae above give a framework to find the first two moments of the excess

volumes. However, in many performance questions the interested part is the tail of the dis-

tributions, which generally is much harder to obtain. In the following the aim is also to in-

clude the volume in the analysis and to do so we must also include the volume in the distri-

bution (3.21). In the succeeding we shall assume that the bit rate process is continuous in

time and space, and we limit ourselves to consider the most interesting case, the excess vol-

umes when the process is above the capacity level . (A similar analysis is possible to per-

form for the normal case when the process is below the level .) We let

Et 0lim dB t[ ] E dB t

2[ ]

t 0lim 2a= =

Ct

2n

-----

n lim C t( )

t 0lim C= =

Et 0lim dB t[ ] 2 C Cd

C =

=

t( ) 1 at o t2

( )+= t a

a

2---

C C a

2---d

C =

0( )=

C

C


40/278

- 26 -

be the excess volume up to a certain time , and we assume that the fol-

lowing time dependant probability distribution is known:

for , (3.50)

with the corresponding PDF:

(3.51)

Based on these time dependant functions we will first derive the joint PDF of the first pas-

sage time and the corresponding excess volume . We let

be the joint CDF and

the corresponding joint PDF. The event down crossing in the interval , that is

, is equivalent with the event . This event

may be written as the difference (see figure

3.3).

From the definition of and we have

At B C( ) d

0

= t

FCx y z t , , ,( ) P Bt y A,> t z Inf 0 t,( ) B C B0 x=>,>{ }= x C y C

fCx y z t , , ,( ) y z

2

FCx y z t , , ,( )=

Tx

Ax

ATxB C( ) d

0

Tx

= =

FA

xT

x x z t , ,( ) P Ax

z T,>x

t>( )=A

xT

x x z t , ,( )t z

2

FA

xT

x x z t , ,( )=

t t dt +,( )

Tx

t t dt +,( ) Inf 0 t,( ) B C Bt dt+ C,>{ }

Inf 0 t,( ) B C>{ } Inf 0 t dt+,( ) B C>{ }

Figure 3.3: The excess time and excess volume and down crossing instances of thebit rate process.

Bt

time

level C

t

Tx

Ax

t=0 t+dt

Tx

Ax

P Ax

z z dz+,( ) T,x

t t dt +,( )( ) P At z z dz+,( ) Inf 0 t,( ) B C Bt dt+ C,>,( )= =


41/278

- 27 -

The first part of this expression is given by

. The second part is some

more difficult to obtain (expressed in terms of the function ) because it in-

volves the changes in the volume in the interval . Some informally we have

which implies , so to have

we must have . Conditioning onand integrating we then get:

By expanding for small we finally get:

(3.52)

The main restrictions we put on the bit rate process to make the derivation above cor-

rect are mainly that the sample paths of must be continuous. With this assumption we

have and due to the

continuity of the sample paths the last integral will of order where can be made arbi-

trarily small (depending on ). Unfortunately the method fails if the sample paths of the

process contain some kind of jumps. In this case the evolution of the volume (in time) will

depend on the bit rate both before and after the jumps. This is in contrast to the results de-

rived for the excess times in section 3.2.2 where no particular assumption is made about the

continuity of the sample paths. (In chapter 5 we consider the case where the bit rate proc-ess is a semi-Markov process and therefore containing jumps.)

To find the time dependent CDF (3.50) and PDF (3.51) for specific models is of cause the

hard part to find explicit expressions of the joint excess distributions. However, these func-

tions must have some specific initial and boundary conditions to make the proposed de-

scription meaningful. Firstly, the initial conditions obtained by letting in the defini-

tion of , imply that:

P At z z dz+,( ) Inf 0 t,( ) B C>,( ) P At z z dz+,( ) Inf 0 t dt+,( ) B C>,( )

P At z z dz+,( ) Inf 0 t,( ) B C>,( ) fCx( y z, , t), yd zd

y C=

=

Cx( y z, , t),

t t dt +,( )

dA t Bt C( )dt= At dt+ At Bt dt+ C( )dt+= At z z dz+,( )

At dt+ z Bt dt+ C( )dt+ z Bt dt+ C( )dt dz+ +,( )Bt dt+ y=

P At z z dz+,( ) Inf 0 t dt+,( ) B C>,( ) fCx( y z y C ( ) td+, , t td+ ), yd zd

y C=

=

dt

TxA

x x z t , ,( ) y C( )z

fCt

fC+ yd

y C=

=

Bt

Bt

At dt+ A t B C( ) d

t=

t dt+

Bt dt+ C( ) t Bt dt+ B( ) d t=

t dt+

+d= =

dt

dt

t 0

fC


42/278


43/278

- 29 -

where is the double LST (Laplace-Stieltjes Transform) of in the variable

and defined by:

(3.59)

3.3.3 Joint distribution of the excess times and excess volumes

We shall now proceed with a quite similar analysis as above to get the joint PDF of the pair

, bearing in mind that this pair of variable is determined through a second order

limiting procedure, that is we both require up crossing in a small interval and also

down crossing in a second small interval . Before entering the analysis we de-

fine the following joint density:

, then by (3.50)

and (3.51)

. (3.60)

We let be the joint CDF of the pair and

the corresponding joint density function. Some informal we

have:

(3.61)

provided that the limit exists. By using a similar approach as we applied for the conditionalexcess times and excess volumes it is possible to expand the nominator to second order for

small and . These rather technical details are placed in Appendix A where we find:

Cx y s, , ,( ) fC z

t

Cx y s, , ,( ) e

t 0=

st z

fCx y z t , , ,( ) zd td

z 0=

=

TkAk,( )

0 dt1,( )

t t dt 2+,( )

gCx y z t , , ,( )dxdydz =

P B0 x x dx+,( ) B, t y y dy+,( ) A, t z z dz+,( ) Inf 0 t,( ) B C>,( )

gCx y z t , , ,( ) x( )fCx y z t , , ,( )=

FAkTkz t,( ) P Ak z Tk,> t>( )= TkAk,( )

fAkTkz t,( )

t z

2

FAkTkz t,( )=

fAkTkz t,( )dzdt2

P Ak z z dz+,( ) Tk t t dt 2+,( )and upcrossing in 0 dt1,( ),( )

P upcrossing in 0 dt1,( )( )-----------------------------------------------------------------------------------------------------------------------------------------------------

dt1 0lim= =

P At z z dz+,( ) B, 0 C Inf dt1 t,( )B C Bt dt2+

C,>,( )

P B0 C Bdt1C>,( )

----------------------------------------------------------------------------------------------------------------------------------------------dt1 0lim

dt1 dt2

P At z z dz+,( ) B, 0 C Inf dt1 t,( ) B C Bt dt2+ C,>,( ) =


44/278

- 30 -

for small and

.

If is possible to expand the denominator in (3.61) to first order for small the joint den-

sity function for will be given by:

(3.62)

An alternative way of writing (3.62) is found by applying the conditional PDF

given by (3.52) and then define:

. (3.63)

giving the following alternative way of writing :

(3.64)

Unfortunately the restriction we have put on the bit rate process will somehow limit the

usefulness of the last derived formula. As for the conditional down crossing the analysis is

limited by the assumption that the sample paths of the bit rate process have continuous sam-

ple paths. Secondly, and perhaps more restrictive, is the claim on behaviour of the excess

probability for small . As stated in section 3.3.1 it is sufficient that the autocorrela-

tion function is on the form for small where is a positive con-

stant.

The functional relation given by (3.63) and (3.64) is on the same form as (3.52) and there-

fore it is easy to write down the corresponding LST. Before doing this we must examine the

conditional PDF at the boundaries and since these specific val-

ues will be included in the transform. Some informally it is clear that if lies in the inter-

val for the volume must be positive and therefore we must have

for . Then lies in the interval for small the condition-

al density function of the volume given must have an impulse like shape. We

must therefore have for some function .

y C( ) x C( )z

2

2

gCx y 2C+( )

z t

2

gC

t2

2

gC+ +

x C=

xd y zd t1d t2ddy C=

dt1

dt2

dt1

TkAk,( )

AkTkz t,( )

1

C

------ y C( ) x C( )

z

2

2

gCx y 2C+( )

z t

2

gC

t2

2

gC+ +

xd yd

x C=

y C=

=

Ax

Tx x z t , ,( )

hCx z t , ,( ) x( )fAxTx x z t , ,( )=

AkTkz t,( )

AkTkz t,( )

1

C------ x C( )

z

hCt

hC+ xd

x C=

=

C t( ) t

t( ) 1 at2

o t2

( )+= t a

AxTx x 0 t, ,( ) AxTx x z 0, ,( )

Tx

t t dt +,( ) t 0> Ax

fA

xT

x x 0 t, ,( ) 0= t 0> Tx

0 dt,( ) dt

Ax

Tx

Ax

Tx x z 0, ,( ) h x( ) z( )= h x( )


45/278

- 31 -

If we let denote the double LST for the stochastic variables

we get from the equation (3.64):

(3.65)

Since we claim it follows that . Then by insert-

ing for from (3.58) in (3.65) we finally end up with the following expres-

sion:

(3.66)

where is the double LST of defined in (3.59).

We are pleased to note that formula (3.66) contains all the previous formulae for the

first and second order moments of the excess times and excess volumes. By direct dif-

ferentiation it is easy to verify that the first and second order moments of the excess

time and excess volume coincide with the formulae, (3.30) and (3.32), (3.40). We may

also find the correlation between the excess time and excess volume by first evaluating:

(3.67)

(3.68)

We may also find the LSTs for the marginal distributions

and from the result

above. For the excess time we get:

(3.69)

TkAk

s ,( ) E esTk Ak

[ ]=

Tk Ak,( )

TkAk

s,( )1

C------

x C=

x( ) h x( ) x C( ) s+( )fT

xA

x x s , ,( )[ ]dx=

fTkAk 0 0,( ) 1=

x C=

x( )h x( )dx C=

TxAx x s, ,( )

fTkAk s,( )1

C------- C s

x C=

x( )dx x C( )x C=

x( )dx x C( ) s+( )y C=

x C=

+ y C( ) s+( ) x( )fCx y s , , ,( )dydx

=

fCx y s, , ,( ) fCx y z t , , ,( )

E TkAk[ ] s

2

fTkAk

0 0,( )

2 C t( )

t 0=

dt

C----------------------------------= =

C t( ) x y 2C+( )

z 0=

x( )

x C=

x C=

fCx y z t , , ,( )dxdydt=

fTk s( ) E esTk[ ] fTkAk 0 s,( )= =

Ak

( ) E eAk[ ] fTkAk 0,( )= =

Tk

s( )1

C------- C s

x C=

x( )dx s2

x( )fCx y 0 s, , ,( )

y C=

x C=

dydx+

=


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- 32 -

and for the excess volume the similar result is:

(3.70)

3.4 Some concluding remarks

In this chapter we have shown that it is possible to relate many important performance

characteristics to some basic fundamental properties which essential is described by the

joint probability (3.50). This function will therefore be a natural starting point when

studying specific models. The main hindrance in finding this probability is of cause the

claim that the process shall have no down crossing in the interval up to the given time. Unless for some few specific type of processes it is extremely difficult to obtain closed

form expression for probabilities involving the minimum and maxi-

mum . By relaxing on the assumption on the minimum could hope-

fully give reasonable accurate results for small but will surely give inaccurate results

for larger values of . One possible way to improve such an approach is to divide the

interval into say points and then calculate the

-dimensional probability:

for , (3.71)

A natural choice will be to divide the interval by equally spacing; that is (for

). We must, however, be aware that this partition (of the interval) not neces-

sary will lead to a decreasing sequence in as the partition in section 3.2.1 would have

given.

In principle, it could thereby be possible to obtain approximation to nth order of the ex-

cess probabilities by using the corresponding approximative function for (3.50) defined by

(3.72)

as starting point for an approximative analysis. In the next chapter in section 4.3 we have

analysed the -point approximation for Gaussian processes and developed methods that en-

able obtaining the approximative distributions for the excess times for up to , and

where we also have compared with corresponding exact results for the O-U process. As dis-

cussed in chapter 4 the differences between the exact and approximations are pronounced,

also when the numbers of points are taken as high as . This example indicates that

the convergence by (3.71) and (3.72) may be quit slow.

Ak

( )1

C------- C x C( )

x C=

x( )dx 2

x C( ) y C( ) x( )fCx y 0, , ,( )

y C=

x C=

dydx+

=

t

mt Inf 0 t,( ) B=

Mt Su p 0 t,( ) B=

t

t

0 t,( ) m t0 0 t1 tm 1 tm< < < < t= =

m 1+( )

GCm

x y z t , , ,( ) P B0 x> Bt1 C> ... Btm 1, C> B, , , t y A,> t z>{ }= x C y C

tii

m----t=

i 0 1 m, , ,=

m 2n

FCn

x y z t , , ,( )x

GCn

x y z t , , ,( )( ) x( )=

m

m 6=

m 6=


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4

4Transient behaviour of Gaussian traffic modelsthrough level crossing

4.1 Introduction

Gaussian modelling has been widely used as a powerful and successful tool in applied sci-

ence. The different application areas vary from statistical communication theory to differ-

ent areas in physics. Traditionally, in traffic theory, where the models usually have been of

discrete nature, the continuous state models have been devoted less attention. Quite recent-

ly however, different Gaussian models have turned out to constitute an important analytical

framework to describe newly observed phenomena in traffic streams. For example the self-

similar behaviour observed for some type of Internet traffic may be described and analysed

by applying fractional Brownian motion as arrival process. Many interesting new results for

self-similar traffic may be found in the bookSelf-similar Network Traffic and performance

Evaluation edited by K. Park and W. Willinger [Park00].

For Gaussian models level crossing have been studied for a rather long period. Best known

are perhaps the early works of O. C. Rice, [Rice45] and [Rice48], where the famous Rices

formula on the crossing rate for Gaussian processes is given. He also gives some prelimi-

nary results on the distribution between two successive zeros. These results have been ex-

tended by J. McFadden and M. S. Longuet-Higgens in later works [McFa56], [McFa58] and

[Long58], [Long62] where different approximations to get the distribution between succes-

sive zeros are discussed. These papers are all based on problems within the field of random

noise and are not direct applicable to traffic models where we are interested in deviations of

the processes having small probabilities rather than variations around the mean value.

There is one main concern when applying Gaussian models to describe network traffic. This

is due to the irregular sample paths for such models. It turns out that the autocorrelationmust have specific behaviour near the origin to have finite up and down crossing intensities

[Lead83]. This limitation fits rather badly with the possibility of having so called long-

range dependence where the autocorrelation behaves as

as for (4.1)

where as the requirement of having finite up and down crossing intensities requires

[Lead83]:

t( ) ct2H-2

t 1 2 H 1<


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- 34 -

as (4.2)

If we for instance consider an autocorrelation function on the form ,

then (4.1) is fulfilled for but to get (4.2) this requires which gives a

process that is not long-range dependent. In a paper by A. Barbe [Barb92] a method is de-

scribed to relax condition (4.2). This can be done by considering a sampled version of the

process (and let the process be linear between samples) and only counting the crossings of

the sampled process. For the sampled process the crossing intensities will be finite. There is

however, a problem to choose the appropriate size of the sampling interval. We shall use

the first passage time and the corresponding volume as an alternative measure when thecrossing intensities are infinite since these distributions are possible to obtain independent

of the crossing rates.

4.2 Gaussian traffic models

In the following we shall consider a stationary Gaussian (normal) random process

with Mean value and Standard deviation , and with autocorrelation function . For

a given capacity level we also let be the excess volume. In the suc-

ceeding subsections we shall work with scaled variables defined by: and

where we also have introduced the scaled capacity by

. We let also be the normalized arrived volume

of the process. (Below we shall omit the but remember that through the rest of this

chapter we are working with normalized variables as defined above.)

For Gaussian (normal) processes it is possible to relax the requirements on the autocorrela-

tion function to secure that the process has continuous sample path [Lead83]. It is suffi-cient that the autocorrelation is bounded by for some and .

For a (standard) stationary Gaussian (normal) process it is well known that a necessary and

sufficient condition for the up and down crossing intensity

(4.3)

to exists and that the limit is finite, is that the autocorrelation takes the specific form:

t( ) 1 at2

t 0

t( )1

1 at2 2H

+---------------------------=

1 2 H 1< < H 0=

Bt{ }

m t( )

C At B C( ) d

0

t

=

Bt

Bt m

---------------=

AtAt

----- B C( ) d

0

t

= =

CC m

--------------= At

B d

0

At Ct+= =

1 t( ) at

a 0> 0>

C Cuc

t 0lim t( )

P B0 C Bt<


49/278


50/278

- 36 -

and and (4.8)

and (4.9)

where we also have used the following expressions for the integrals:

and .

We may also find the overall information loss given by (3.34) as:

(4.10)

and the information loss in an overload period given by (3.35) as:

(4.11)

Note that the information losses do not depend on the parameter but only on the capaci-

ty level . Asymptotics for large values of are easily found by using asymptotic expan-

sion for the normal integral for large , cf. [Abra70] page 932 formula 26.2.10, giv-ing:

and and further (4.12)

and (4.13)

Thus, even though the overall information losses may be well limited, the losses in an over-

load period will be significant as shown by the asymptotics (4.13).

E Tk[ ]a---

C( ) C( )------------= E Sk[ ]

a---

1 C( ) C( )

------------------=

E Ak[ ]a--- 1 C

C( ) C( )------------

= E Vk[ ]a---

C

C( )------------ 1 C

C( ) C( )------------

=

x( ) xd

C

C( ) C C( )= 1 x( )( ) xd

C

C C( ) C C( )( )=

ploss

ploss C( ) C C( )=

qloss

qloss C( ) C( )------------ C=

a

C C

C( ) C

E Tk[ ]a---

1

C---- E Ak[ ]

a---

1

C2

------

ploss C( )

C2

------------ qloss1

C----


51/278

- 37 -

In the figure 4.1 we have plotted the overall information loss probability based on equation

(4.10) (left and marked exact) and the loss probability in overload periods based on equa-

tion (4.11) (right and marked exact) together with the asymptotics given by (4.13). If we

for instance would like to keep the losses in the range 10 -2-10-4 this gives the interesting

parameter value of the scaled capacity to be in the range 2.0-3.2.

4.3 Then-point approximation for Gaussian processes

Unless for some few cases it is difficult to obtain exact expressions for the excess probabili-

ties involving the minimum and maximum of the

process. In chapter 3 we saw that the -point approximation (obtained by continuously bi-

secting each interval) lead to approximations that were monotone and therefore had nice

properties to secure convergence of the corresponding approximations to the exact proba-

bilities. However, due to the difficulties to calculate multinormal integrals we shall apply

the -point approximation by dividing the interval into sub-intervals of equal lengths,

(where we keep in mind that the -point approximation coincides with the bisecting meth-

od for .)

More generally, and due to the property of a stationary Gaussian random process, wehave that for every sequence of succeeding points, say

, that the ensemble is Multivariate

Gaussian distributed with zero mean and Covariance matrix given by

(4.14)

Figure 4.1: Logarithmic plot of the overall loss probability (left) and loss proba-bility in overload periods (right) as function of the capacity.

1 2 3 4 5

-4

-3

-2

-1

0

1 2 3 4 5

-0.8

-0.6

-0.4

-0.2

0

capacity C capacity C

asymptotic

exact

asymptotic

exact

ploss

qloss

mt Inf 0 t,( ) B= Mt Su p 0 t,( ) B=

2n

n

n

n 2 4 8, ,=

n

t1 0 t2 tn 1 tn< < < < t= = Bt1 Bt2 , , Btn,{ }

M ij( )=

ij E BtiBtj[ ] ti tj( )= =


52/278

- 38 -

The joint distribution function for a Multivariate normal process is well known and is ex-

pressed in terms of the inverse of the covariance matrix . (See (B.1) in

Appendix B.) With the notation introduced in Appendix B (see equation (B.2)) we may

express the -point approximation of the excess probability as:

(4.15)

where is given by (4.14) and the vector giving the integration limits and

where is the scaled capacity. We shall apply the results derived in Appendix B to get

more explicit expressions for the excess time distribution. By theorem B.3 in Appendix B

(equation (B.65)) this integral may be written as:

(4.16)

where and is given as and in theorem B.2 and corollary B.1 in Ap-

pendix B and are explicitly given below in (4.30) and (4.31) but with replacing with

for all , and further is the standard normal integral.

The main achievements by writing the integral (4.15) of the form (4.16) is the fact that the

dimension of the integral in the integrand is reduced to which means that the num-bers of possible numerical integrations is reduced by . Continuing this process we obtain

the remarkable result which shows that it is possible to calculate the -dimensional multi-

normal integral by only performing (if is even) or (if is odd) successive

numerical integrations. This very particular property of the multinormal integrals is widely

applied and enable us to calculate multinormal probabilities of dimension seven by perform-

ing only three numerical integrations.

In the following we shall apply the -point approximation for to find approximations of the

various excess distributions defined in chapter 3. A natural choice of the points would be

to take them equally spaced, that is:

for (4.17)

which gives for . (4.18)

4.3.1 n-point approximation of the distribution of excess times and first pas-

M1

Mij1

( )= M

n

Cn

t( ) I n CM, ,( )=

M C C ...,C,( )=

C

Cn

t( ) C( )n

kle

C2

1 kl+-------------------

2 1 kl( )2

--------------------------------------I n 2 Ck l,

Mk l,

, ,( ) d

0=

1

1 k l n

Ci1

i 2 ... n, ,=

FT

x

nt( )

t 0lim I n 1 Cx

1M

1, ,( )

t 0lim 1= =

x C= Ci1

0

i 2 ... n, ,= ij1

1

FT

xn t( )

t 0lim 0( ) 1

2---= = t( )

M1

0( )

I n 1 0 M1

0( ), ,( ) 0

n


54/278

- 40 -

(4.23)

(4.24)

The corresponding PDF may be found by differentiating the integral (4.19), giving:

(4.25)

By first differentiating with respect to the time, applying corollary B.4 and theorem B.6

(Appendix B) we find the following expression for the -point approximation of the PDF

of the first passage time :

(4.26)

where the and are integrals of type (B.50) and may be writ-

ten as products of bi- and tri-normal distributions and standard multinormal integrals of di-mension and by applying conditional distributions as given in Appendix B by

equation (B.51).

To find the -point approximation of the excess time distribution we first examine the be-

haviour of for small . By the general results in chapter 3 equation (3.18) we find

the following bounds for :

(4.27)

It follows now since that . Then

by letting in the last inequality we find that if is finite

then . We therefore conclude that the -point approximation

will lead to the correct up crossing intensity for all values of and we find the approxima-

tive distribution that may be defined as in chapter 3 by:

FT

x

2t( )

C x t( )

1 t( )2

-------------------------

=

FT

x

3t( ) I 2

C x t 2( )

1 t 2( )2

--------------------------------C x t( )

1 t( )2

------------------------- t 2( )1 t( )

1 t( )+( ) 1 t 2( )2

( )----------------------------------------------------------, , ,

=

Tx

nt( )

1

f1 x( )------------

t x

2

I n Cx M, ,( )1

f1 x( )------------

x

tI

n Cx M, ,( )

= =

n

Tx

fT

x

nt( )

1

f1 x( )------------

1kd

td-----------

x C1k

1 1k2

---------------------I

1 k,n Cx M, ,( )

kld

td----------

1kd

td-----------

1l 1kkl

1 1k2

-----------------------------

1ld

td----------

1k 1lkl

1 1l2

----------------------------- I

1 k l, ,n Cx M, ,( )

2 k l n


55/278

- 41 -

(4.28)

We have . By applying theorem B.2 and corollary B.3 (in

Appendix B) we get:

where (4.29)

and where the -vector is given by:

for , (4.30)

and the correlation matrix is given by:

(4.31)

for ,

The specific choice of the points makes it possible to rewrite equation (4.29) by group-

ing factors for which and this somehow simplifies the expression for :

(4.32)

(4.32) represents the limit of how far it is possible to analyse the -point approximation for

general correlation function. The main difficulty to get a proper excess distribution is the

behaviour of the approximation for small due to the square root in the denominator. To

examine the behaviour for small we shall examine

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